The Growth and Structure of Dark Matter Haloes

arXiv:astro-ph/0204108v27Oct2002Mon.Not.R.Astron.Soc.000,000–000(2000)Printed1February2008(MNLATEXstylefilev1.4)TheGrowthandStructureofDarkMatterHaloesD.H.Zhao1,2,H.J.Mo2,Y.P.Jing1,2,G.B¨orner21ShanghaiAstronomicalObservatory,thePartnerGroupofMPIf¨urAstrophysik,NandanRoad80,Shanghai200030,China2Max-PlanckInstitutf¨urAstrophysik,Karl-Schwarzschild-Strasse1,85748Garching,[e-mail],[e-mail]@center.shao.ac.cn,[e-mail]........Received.......;inoriginalform.......ABSTRACTInthispaper,weanalyseindetailthemass-accretionhistoriesandstruc-turalpropertiesofdarkhaloesinhigh-resolutionN-bodysimulations.WemodelthedensitydistributioninindividualhaloesusingtheNFWprofile.Foragivenhalo,thereisatightcorrelationbetweenitsinnerscaleradiusrsandthemasswithinit,Ms,forallitsmainprogenitors.Usingthiscorrela-tion,onecanpredictquitewellthestructuralpropertiesofadarkhaloatanytimeinitshistoryfromitsmassaccretionhistory,implyingthatthestructurepropertiesandthemassaccretionhistoryarecloselycorrelated.Thepredictedgrowingrateofconcentrationcwithtimetendstoincreasewithdecreasingmassaccretionrate.Thebuild-upofdarkhaloesinCDMmodelsgenerallyconsistsofanearlyphaseoffastaccretion[wherethehalomassMhincreaseswithtimemuchfasterthantheexpansionrateoftheuniverse]andalatephaseofslowaccretion[whereMhincreaseswithtimeapproximatelyastheexpansionrate].Thesetwophasesareseparatedatatimewhenc∼4andthetypicalbindingenergyofthehaloisapproximatelyequaltothatofasingularisothermalspherewiththesamecircularvelocity.Haloesinthetwoaccretionphasesshowsystematicallydifferentproperties,forexample,thecircularve-locityvhincreasesrapidlywithtimeinthefastaccretionphasebutremainalmostconstantintheslowaccretionphase,theinnerpropertiesofahalo,suchasrsandMsincreaserapidlywithtimeinthefastaccretionphasebutchangeonlyslowlyintheslowaccretionphase,theinnercircularvelocityvsisapproximatelyequaltovhinthefastaccretionphasebutislargerintheslowaccretionphase.Thepotentialwellassociatedwithahaloisbuiltupmainlyc2000RAS2Zhaoetal.inthefastaccretionphase,whilealargeamountofmasscanbeaccretedintheslowaccretionphasewithoutchangingsignificantlythepotentialwell.Wediscussourresultsinconnectiontotheformationofdarkhaloesandgalaxiesinhierarchicalmodels.Keywords:galaxies:formation-galaxies:clusters-large-scalestructure-cosmology:theory-darkmatter1INTRODUCTIONTheColdDarkMatter(CDM)cosmogony(Peebles1982;Blumenthaletal.1984;Davisetal.1985)hasnowbeenprovedtobearemarkablysuccessfulframeworkforunderstandingthestructureformationintheuniverse.IntheCDMscenario,akeyconceptinthebuild-upofstructureistheformationofdarkmatterhaloes,whicharequasi-equilibriumsystemsofdarkmatterparticlesformedthroughnon-lineargravitationalcollapse.InahierarchicalscenarioliketheCDM,mostmassintheUniverseatanytimeisindarkhaloes;galaxiesandotherluminousobjectsareassumedtoformbycoolingandcondensationofbaryonswithinthesehaloes(White&Rees1978).Inthiscase,adetailedunderstandingoftheformationandstructureofdarkmatterhaloesisoffundamentalimportanceforpredictingthepropertiesofluminousobjects.Oneofthemostimportantpropertiesofthehalopopulationistheirdensityprofiles.BasedonN-bodysimulations,Navarro,Frenk,&White(1997;hereafterNFW),foundthatCDMhaloescanbefittedbyatwo-parameterprofile:ρ(r)=4ρs(r/rs)(1+r/rs)2,(1)wherersisacharacteristic“inner”radiusatwhichthelogarithmicdensityslopeis−2,andρsisthedensityatrs.Ahaloisoftendefinedsothatthemeandensityρhwithinthehaloradiusrhisaconstant(Δh,willbedefinedinSection2)timesthemeandensityoftheuniverse(¯ρ)attheredshift(z)inconsideration.ThehalomasscanthenbewrittenasMh≡4π3Δh¯ρr3h.(2)Wedefinethecircularvelocityofahaloasvh=(GMh/rh)1/2,andsoMh=v2hrhG=21/2v3h[ΔhΩ(z)]1/2H(z),(3)whereH(z)istheHubbleconstant,andΩ(z)themassdensityparameter,atredshiftz.NFWc2000RAS,MNRAS000,000–0003introducedanalternativeparameter,theconcentrationparameterc,definedasc≡rh/rs,todescribetheshapeofthehaloprofile.Itistheneasytoshowthatρs=ρhc312[ln(1+c)−c/(1+c)].(4)WedenotethemasswithinrsbyMs,andthecircularvelocityatrsbyvs.ThesequantitiesarerelatedtocandMhasMs=ln2−1/2ln(1+c)−c/(1+c)Mh,v2s=v2hcMsMh.(5)Ingeneral,thestructureofahaloisexpectedtodependnotonlyoncosmologyandpowerspectrum,butalsoonitsformationhistory.Therearethereforeattemptstorelatehaloconcentrationtootherquantitiescharacterizingtheformationofahalo.Intheiroriginalpaper,NFWsuggestedthatthecharacteristicdensityofahalo,ρs,shouldbeaconstant(k)timesthemeandensityoftheuniverse,¯ρ(zf),attheredshiftzf(referredastheformationtimeofthehalobyNFW)whenhalfofthehalo’smasswasfirstinprogenitorsmoremassivethanftimesthehalomass.NFWusedtheextendedPress-Schechterformulatocalculatezfandfoundthattheanti-correlationbetweencandMhobservedintheirsimulationscanbereproducedwithaproperchoiceofthevaluesfortheconstantskandf.Subsequentanalysesshowedthatadditionalcomplexitiesmaybeinvolvedinhalostruc-ture.First,ithasbeenshownthathaloesoffixedmasshavesignificantscatterintheircvalues(Jing2000),althoughthereisameantrendofcwithMh.Ifthistrendisindeedduetoacorrelationbetweenconcentrat